To find the average velocity of the train between minutes \( t = 3 \) and \( t = 7 \), we can use the formula for average velocity over an interval, which is:
\[ \text{Average velocity} = \frac{f(b) - f(a)}{b - a} \]
where \( f(t) \) is the position function, \( a = 3 \), and \( b = 7 \).
First, we need to calculate \( f(3) \) and \( f(7) \):
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Calculate \( f(3) \): \[ f(3) = 90(3^2) + 2(3) \] \[ = 90(9) + 6 \] \[ = 810 + 6 \] \[ = 816 \text{ miles} \]
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Calculate \( f(7) \): \[ f(7) = 90(7^2) + 2(7) \] \[ = 90(49) + 14 \] \[ = 4410 + 14 \] \[ = 4424 \text{ miles} \]
Now, we can find the average velocity:
\[ \text{Average velocity} = \frac{f(7) - f(3)}{7 - 3} \] \[ = \frac{4424 - 816}{4} \] \[ = \frac{3608}{4} \] \[ = 902 \text{ miles per minute} \]
Therefore, the average velocity of the train between minutes 3 and 7 is \( 902 \) miles per minute.